Theorems · Theorem · functional analysis
Submodule.norm_orthogonalProjectionOnto_apply_le
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(K : Submodule 𝕜 E) [inst_3 : K.HasOrthogonalProjection] (v : E), ‖K.orthogonalProjectionOnto v‖ ≤ ‖v‖The orthogonal projection onto a closed subspace is norm non-increasing.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Realstatement · cited by 25,697
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement and proof · cited by 7,192
- Norm.normstatement and proof · cited by 5,413
- ContinuousLinearMapstatement · cited by 5,352
- InnerProductSpacestatement and proof · cited by 3,523
- one_mulproof · cited by 2,841
- RCLikestatement and proof · cited by 2,829
- norm_nonnegproof · cited by 725
- mul_le_mul_of_nonneg_rightproof · cited by 301
Cited by2
Results whose statement or proof uses this declaration.
- Submodule.norm_starProjection_apply_leproof · cited by 1
- Submodule.norm_orthogonalProjection_apply_leproof · cited by 0